MATH1131 Calculus
Chapter 1
Sets, inequalities and functions 1
Sets and elements
◮
A set is a collection of distinct objects.
◮
The objects in a set are called the elements or members of the set.
Example 1.1 ◮
MATH1131 is an element of the set of all mathematical courses offered at UNSW.
◮
UCLA is not a member of the Australian universities.
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Some commonly used sets of numbers N Z Z+ Q R
C
set of all natural numbers 0, 1, 2, . . . set of all integers 0, ±1, ±2, . . . set of all positive integers 1, 2, 3, . . . set of all rational numbers, i.e. those numbers expressible as a ratio of two integers set of all real numbers R may be represented as the number line:
set of all complex numbers
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Example 1.2 We write 2 ∈ N and read 2 is an element of the natural numbers and −
4 ∈ /Z 3
and read −4/3 is not an element of the integers
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The empty set ∅ The empty set or null set, denoted by ∅, is a set which has no members at all.
Example 1.3 ◮
The set of all students doing MATH1131 in 2010 whose ages are below 10 is an empty set.
◮
The set of all real numbers x satisfying x 2 + 1 = 0 is an empty set.
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Two ways to define a set ◮
◮
List all members inside a pair of curly brackets (or braces): {}. Give a rule to determine which members of some previously defined set are to be members of the set being defined.
Example 1.4 ◮
The set of all natural numbers less than 3 can be represented by one of the following ways {0, 1, 2}
◮
or {2, 1, 0}
or {n ∈ N : n < 3}.
The equation x 2 = 1 has two real solutions 1 and −1. The set of all solutions of this equation can be written as {−1, 1} or {x ∈ R : x 2 = 1}.
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Some useful jargon Definition 1.1 (Subset) ◮
◮
Suppose that A and B are two sets. If x ∈ A implies that x ∈ B, then A is called a subset of B. We write A ⊂ B.
If A is a subset of B we can also say that B contains A, and write B ⊃ A.
Example 1.5 N⊂Z
R⊃Q
(N
is a subset of Z)
(R
contains Q).
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Intervals If a < b are two real numbers, we write (a, b) [a, b] (a, b] [a, b)
for
{x ∈ R : a < x < b}
— an open interval
{x ∈ R : a ≤ x ≤ b}
— a closed interval
{x ∈ R : a < x ≤ b}
{x ∈ R : a ≤ x < b}
[a, ∞)
{x ∈ R : a ≤ x }
(a, ∞)
{x ∈ R : a < x }
(−∞, a] (−∞, a)
{x ∈ R : x ≤ a}
{x ∈ R : x < a}
Note ∞ is not a real number. The only sensible relations involving the symbol ∞ and real numbers x and y are −∞ < x
or
y < ∞. 8
Inequalities Theorem 1.1 For x, y, z ∈ R we have
1. if x > y then x + z > y + z 2. if x > y and z > 0 then xz > yz 3. if x > y and z < 0 then xz < yz √ √ 4. if y > x > 0 then y > x > 0 1 1 5. if y > x > 0 then 0 < < y x
Note Note the change of direction of the inequality signs in (3) and (5).
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Solving inequalities There are 2 basic types: ◮ polynomial inequalities ◮ ◮ ◮
◮
linear inequalities quadratic inequalities maybe higher degree inequalities if easy
rational function inequalities — which will usually reduce to polynomial inequalities
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! Example 1.6 Find S = {x ∈ R : 3(4 − x) < 21}.
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! Example 1.7 Find T = {u ∈ R : u 3 − 3u > 2u 2 }.
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! Example 1.8 Find the set of all x ∈ R satisfying 1 1 > . (x − 1)(x − 2) (x − 1)(x − 3)
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Notes ◮ ◮
∪ is read union.
To be precise, when we multiply (x − 1)2 (x − 2)2 (x − 3)2 to the inequality we have to assume that x 6= 1, 2, 3. Otherwise, the inequality becomes equality. The assumption is clearly valid so that the fractions are defined.
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Absolute values Definition 1.2 The absolute value |x| of a real number x is defined by ( x if x ≥ 0 |x| = −x if x < 0.
Theorem 1.2 ◮ ◮ ◮ ◮
| − x| = |x|
|xy| = |x| |y|
|x ± y| ≤ |x| + |y| |x| − |y| ≤ |x − y|
(the triangle inequality) (the circle inequality)
You have to be careful with problems with absolute values! 15
Important facts ◮
If x, y ∈ R then |x − y| is the distance between x and y on the real number line.
◮
If x ∈ R then
√
x 2 = |x|
and |x|2 = x 2 .
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Important facts ◮
If a > 0 then ◮
◮
◮
|x | < a means −a < x < a; equivalently, we can write x ∈ (−a, a) |x − x0 | < a means x0 − a < x < x0 + a; equivalently, we can write x ∈ (x0 − a, x0 + a) |x | > a means (x < −a or x > a).
For examples, see the Calculus Notes – Example 1.3.2.
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Functions, domain and codomain Definition 1.3 A function f :A→B is a rule which assigns every element x ∈ A to exactly one element f (x) ∈ B. The set A is called the domain of the function f , and is often denoted by Dom(f ). The set B is called the codomain of f . Hence, to define a function precisely we need a domain, a codomain and a rule. In this course, the domain and codomain are always sets of real numbers. Note every element x in A must have an assignment y in B, but there may be elements y in B which are not an assignment of any element x in A. 18
If a function is defined by a formula, instead of writing,
Example 1.9 The function f :R→R
defined by f (x) = x 2
we can also write f :R→R
x 7→ f (x) = x 2 .
Example 1.10 The following functions are all different f :R→R
x 7→ x 2
g : [0, ∞) → R
x 7→ x 2
h : [0, ∞) → [0, ∞) x 7→ x 2 .
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A convention ◮
◮
It is emphasised that f (x) is a number, the value of the function at x. It is not a function. √ However, sometimes only the rule is given (e.g., f (x) = x). For rules which operate on real numbers and produce real numbers as output we use the following convention: ◮
◮
The default domain is the largest possible set of x values for which f (x ) ∈ R; The default codomain is the smallest possible set of values of f (x ).
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Range Definition 1.4 The range of a function f : A → B, denoted by Range(f ) or f (A), is the set {y ∈ B : y = f (x) for some x ∈ A}.
Notes ◮
The range is a subset of the codomain. f (A) is a subset of B.
◮
The codomain indicates where we should look for the function values — the range tells us the actual function values obtained.
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! Example 1.11 ◮ ◮ ◮ ◮
f : R → R with x 7→ 1 has f (R) ∈
g : Z → R with n 7→ cos(nπ) has g(Z) ∈
h : R → R with x 7→ cos(xπ) has h(R) ∈
F : (0, 1] → R with x 7→ 1/x has F ((0, 1]) ∈
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Operations on functions Definition 1.5 Suppose that f : D → R and g : D → R are two functions with the same domain D, a subset of R. Then the functions f ± g, f · g and f /g are defined by the rules (f ± g)(x) = f (x) ± g(x) (f · g)(x) = f (x)g(x) f (x) (f /g)(x) = g(x)
∀x ∈ D
∀x ∈ D ∀x ∈ D satisfying g(x) 6= 0.
Thus the domains of these functions are Dom(f ± g) = D
Dom(f · g) = D
Dom(f /g) = {x ∈ D : g(x) 6= 0}. 23
Notes ◮ ◮
The notation ∀x ∈ D is read for all x in D.
Polynomials and rational functions are made using the above ideas.
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Polynomials and rational functions ◮
Polynomials are functions of the form p(x) = a0 + a1 x + a2 x 2 + · · · + an x n where a0 , . . . , an ∈ R. Their domain is R.
◮
Rational functions are functions of the form p(x) q(x) where p and q are polynomials. Their domain is {x ∈ R : q(x) 6= 0}.
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! Example 1.12 Find the domains of the functions defined by these rules 2x f (x) = 2 x +3
x 3 + 5x − 7 and g(x) = 2 . x − 4x + 3
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! Example 1.13 Does the following rule defines a rational function? f (x) = x + 1 −
2 . x2 + 2
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Composition of functions Definition 1.6 Consider two functions: f : C → D and g : A → B. When the range of a function g is a subset of the domain of a function f , we can define the composition f ◦ g : A → D is defined by (f ◦ g)(x) = f g(x) ∀x ∈ A. Note the order of f and g is important!
f g
1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 A 0000000 1111111 0000000 1111111
g
B 111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111
1111 0000 0000 1111 0000 1111 0000 1111 0000 1111
D
C f 28
Example 1.14 If f : R → R and g : R → R are defined by f (x) = sin x
and g(x) = x + 1.
then (f ◦ g)(x) = f g(x) = f (x + 1) = sin(x + 1). (g ◦ f )(x) = g f (x) = g(sin x) = sin x + 1.
Note Compositons do not always exist.
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Example 1.15 Let f : R → R and g : [0, ∞) → R be defined by √ f (x) = x 2 − 1 and g(x) = x. In this case f ◦ g : [0, ∞) → R is defined by √ √ (f ◦ g)(x) = f (g(x)) = f ( x) = ( x)2 − 1 = x − 1. (Note that f ◦ g is different from h : R → R defined by h(x) = x − 1.) Here, however, g ◦ f is not defined. Indeed, if x = 0 then f (x) = −1 and thus g (f (x)) = defined (as a real number).
√
−1 is not
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2 − 1 is not a subset of the We can see that the range of f = x √ domain of g = x.
However, if we restrict the domain of f so that f : D → R where D = {x ∈ R : x 2 − 1 ≥ 0} = {x ∈ R : x ≤ −1 or x ≥ 1} then g ◦ f : D → R is defined by 2
(g ◦ f )(x) = g(x − 1) =
p
x 2 − 1.
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Trigonometric functions (cos t , sin t )
t 1
If t is a real number then (cos t, sin t) are the coordinates of the point on the unit circle x 2 + y 2 = 1 whose distance around the circle in the anticlockwise direction from (1, 0) is t. The angle from the positive x-axis to this point is t radians.
So 2π radians is same angle as 360◦ . From now on always measure angles in radians. From the diagram we have sin t ≤ t
for t ≥ 0. 32
y
sin(t)
1
1
y0
y0
t0
x
t0
x0 1
-1
t p/2
p
3p/2 2p 5p/2 3p 7p/2 4p
-1
-1
t0
cos(t)
1 x0
-1 p/2 p 3p/2 2p 5p/2 3p 7p/2 4p
t
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Domains and ranges of trigonometric functions ◮
The sine and cosine functions Dom(sin) = Dom(cos) = R Range(sin) = Range(cos) = [−1, 1].
◮
Other trigonometric functions tan x =
sin x , cos x
Dom(tan) = {x ∈ R : x 6=
π + k π, k ∈ Z}, 2
Range(tan) = R π + k π, k ∈ Z}, 2 Range(sec) = {x ∈ R : x ≤ −1 or x ≥ 1}
sec x =
1 , cos x
Dom(sec) = {x ∈ R : x 6=
cosec x =
1 , sin x
Dom(cosec) = {x ∈ R : x 6= k π, k ∈ Z},
cot x =
cos x , sin x
Range(cosec) = {x ∈ R : x ≤ −1 or x ≥ 1} Dom(cot) = {x ∈ R : x 6= k π, k ∈ Z}, Range(cot) = R 34
Elementary functions The following are known as elementary functions: ◮
polynomials
◮
the nth root function f (x) = x 1/n where n ∈ Z+
◮
the exponential function f (x) = ex
◮
the natural logarithm function f (x) = ln x
◮
the absolute value function f (x) = |x|
◮
all the trig functions (and their inverses) where you can use high school definitions for the moment!
Any function obtained by combining a finite number of the above functions via +, −, ∗, ÷ and ◦ is also an elementary function.
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! Example 1.16 The function f :R→R
2
such that x 7→ esin
x+3 sin x−1
is an elementary function.
Note In Maple the exponential function is also exp(x).
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Implicitly defined functions The points (x, y) in the plane satisfying the equation x − y2 = 0
(1)
define a curve in the plane. By restricting the values of y, we can now define a function. For example, if we impose the condition y ≤ −1, then we obtain the function √ y = f (x) = − x where f : [1, ∞) → (−∞, −1] We say that f is implicitly defined by (1). Clearly lots of different functions are defined implicitly by (1). In general, an expression involving x and y may define a function y = f (x) if restrictions are placed on y . 37
Example 1.17 The Maple command smartplot(y^3 + 3*x*y^2 + 3*y*x^2 + y + 5=0); can be used to get a picture of the curve y 3 + 3xy 2 + 3x 2 y + y + 5 = 0.
(2)
Challenge: Prove mathematically that (2) defines function y = g(x) with R as its domain.
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Continuous functions A primitive, but vaguely useful, notion of continuity is “a function is continuous if its graph can be drawn without lifting the pencil off the page” Better is “a function is continuous if small changes in the input variable produce small changes in the function value” However, a lot of work is required to convert this to serious mathematics. To make efficient progress in MATH1131 we initially take as a fact that the elementary functions are continuous on their domain of definition. This procedure then allows us to draw diagrams to help understand the important concept of a limit. It also justifies many of the calculations of limits you did in the revision examples.
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